Homepage of Dr Iwan Jensen MASCOS Senior Research Fellow Department of Mathematics and Statistics

Series and other data related to the five-particle contribution χ⁽⁵⁾ to the square lattice Ising model susceptibility.

First a bit of notation. The `natural' variables are s=sinh(2J/kT) and w=s/(2(1+s²))

Some exact results for n-particle contributions to the susceptibility

χ(1) = 2w/(1-4w)
ODE for χ(3) in variable w.
First column is the order k of the derivative, the second column is j and the third column is the j'th coefficient pk,j in the polynomial Pk(w) multiplying the k'th derivative.
ODE for χ(5) in variable w mod the prime 32749.
For χ(5) we express the solution in terms of a linear ODE with polynomial coefficients but using the diffential operator (wd/dw). The solution is of order n=56 with the degree of the polynomials equal 129. The data is organised as a list of lists with the first list being the coefficients of Pn(w), the second list being the coefficients of Pn-1(w) etc. etc. and the last list is the coefficients of P0(w)

 

Some exact series mod various primes.

For the prime 32749 the series has some 10000 terms
while for the remaining primes the series has some 5600 terms.

Series for χ(3) in w mod the primes:     32749     32719     32717     32713

Series for χ(5) in w mod the primes:     32749     32719     32717     32713

 

Some exact results for factors occuring in the order 29 differential operator L₂₉ annihilating the series   Φ⁽⁵⁾=χ⁽⁵⁾-χ⁽³⁾/2+χ⁽¹⁾/120

First we have the operator L₁₁ in exact arithmetic. This operator has the factorization:





The multiplications are done as differential operators and the addition is the direct sum of operators.
In terms of commands from the Maple package DEtools we have:





 


Next we have the operator L₅ mod 32749. This is the minimal order ODE for this operator.
A solution to this ODE is given by







where E and K denote the complete elliptic integrals







while P_4-i,i are polynomials in w with coefficients known modulo the prime 32749 and of degree 200, 202, 204, 204 and 204, respectively. Click here to download these polynomials .



Here we have the operator L₁₃ mod 32749. This is a non-minimal order ODE (order 19) for this operator.

 

In our paper J. Phys A 43 195205 (2010) we calculated the ODE in exact arithmetic. Below are some of main results.

Exact χ⁽⁵⁾ to order 8000, Exact L₂₉ operator, Exact L₂₄ operator, Exact L₁₂(right) operator, and sKE polynomials.

Exact L₁₁ operator and `reduced' L₁₃ polynomials Q_k. The polynomials occuring in L₁₃ are P_k=Q_k*PA^k, where PA is the apparent polynomial of L₁₁

Since publication one of us (Bernie Nickel) has found several exact relations between the constants at the ferromagnetic point given in the arrays C₀ and C₁ of equation (B.4). These relations can be found here